Satisfiability Games for Branching-Time Logics

The satisfiability problem for branching-time temporal logics like CTL*, CTL and CTL+ has important applications in program specification and verification. Their computational complexities are known: CTL* and CTL+ are complete for doubly exponential time, CTL is complete for single exponential time. Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems. In this paper we present a uniform game-theoretic framework for the satisfiability problem of these branching-time temporal logics. We define satisfiability games for the full branching-time temporal logic CTL* using a high-level definition of winning condition that captures the essence of well-foundedness of least fixpoint unfoldings. These winning conditions form formal languages of \omega-words. We analyse which kinds of deterministic {\omega}-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or B\"uchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics. This approach provides a uniform, yet complexity-theoretically optimal treatment of satisfiability for branching-time temporal logics. It separates the use of temporal logic machinery from the use of automata thus preserving a syntactical relationship between the input formula and the object that represents satisfiability, i.e. a winning strategy in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner closure of the input formula only. Last but not least, the games presented here come with an attempt at providing tool support for the satisfiability problem of complex branching-time logics like CTL* and CTL+

Visibly Pushdown Modular Games

Games on recursive game graphs can be used to reason about the control flow of sequential programs with recursion. In games over recursive game graphs, the most natural notion of strategy is the modular strategy, i.e., a strategy that is local to a module and is oblivious to previous module invocations, and thus does not depend on the context of invocation. In this work, we study for the first time modular strategies with respect to winning conditions that can be expressed by a pushdown automaton. We show that such games are undecidable in general, and become decidable for visibly pushdown automata specifications. Our solution relies on a reduction to modular games with finite-state automata winning conditions, which are known in the literature. We carefully characterize the computational complexity of the considered decision problem. In particular, we show that modular games with a universal Buchi or co Buchi visibly pushdown winning condition are EXPTIME-complete, and when the winning condition is given by a CARET or NWTL temporal logic formula the problem is 2EXPTIME-complete, and it remains 2EXPTIME-hard even for simple fragments of these logics. As a further contribution, we present a different solution for modular games with finite-state automata winning condition that runs faster than known solutions for large specifications and many exits.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Reasoning about Dependence, Preference and Coalitional Power

This paper presents a logic of preference and functional dependence (LPFD) and its hybrid extension (HLPFD), both of whose sound and strongly complete axiomatization are provided. The decidability of LPFD is also proved. The application of LPFD and HLPFD to modelling cooperative games in strategic and coalitional forms is explored. The resulted framework provides a unified view on Nash equilibrium, Pareto optimality and the core. The philosophical relevance of these game-theoretical notions to discussions of collective agency is made explicit. Some key connections with other logics are also revealed, for example, the coalition logic, the logic functional dependence and the logic of ceteris paribus preference

Parameterized Linear Temporal Logics Meet Costs: Still not Costlier than LTL

We continue the investigation of parameterized extensions of Linear Temporal Logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for Parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for Parametric LDL (PLDL), which extends PLTL to be able to express all omega-regular properties. Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

Games for Modal and Temporal Logics

Every logic comes with several decision problems. One of them is the model checking problem: does a given structure satisfy a given formula? Another is the satisfiability problem: for a given formula, is there a structure fulfilling it? For modal and temporal logics; tableaux, automata and games are commonly accepted as helpful techniques that solve these problems. The fact that these logics possess the tree model property makes tableau structures suitable for these tasks. On the other hand, starting with Büchi's work, intimate connections between these logics and automata have been found. A formula can describe an automaton's behaviour, and automata are constructed to accept exactly the word or tree models of a formula. In recent years the use of games has become more popular. There, an existential and a universal player play on a formula (and a structure) to decide whether the formula is satisfiable, resp. satisfied. The logical problem at hand is then characterised by the question of whether or not the existential player has a winning strategy for the game. These three methodologies are closely related. For example the non-emptiness test for an alternating automaton is nothing more than a 2-player game, while winning strategies for games are very similar to tableaux. Game-theoretic characterisations of logical problems give rise to an interactive semantics for the underlying logics. This is particularly useful in the specification and verification of concurrent systems where games can be used to generate counterexamples to failing properties in a very natural way. We start by defining simple model checking games for Propositional Dynamic Logic, PDL, in Chapter 4. These allow model checking for PDL in linear running time. In fact, they can be obtained from existing model checking games for the alternating free µ-calculus. However, we include them here because of their usefulness in proving correctness of the satisfiability games for PDL later on. Their winning strategies are history-free. Chapter 5 contains model checking games for branching time logics. Beginning with the Full Branching Time Logic CTL* we introduce the notion of a focus game. Its key idea is to equip players with a tool that highlights a particular formula in a set of formulas. The winning conditions for these games consider the players' behaviours regarding the change of the focus. This proves to be useful in capturing the regeneration of least and greatest fixed point constructs in CTL*. Deciding the winner of these games can be done using space which is polynomial in the size of the input. Their winning strategies are history-free, too. We also show that model checking games for CTL+ arise from those for CTL* by disregarding the focus. This does not affect the polynomial space complexity. These can be further optimised to obtain model checking games for the Computation Tree Logic CTL which coincide with the model checking games for the alternating free µ-calculus applied to formulas translated from CTL into it. This optimisation improves the games' computational complexity, too. As in the PDL case, deciding the winner of such a game can be done in linear running time. The winning strategies remain history-free. Focus games are also used to give game-based accounts of the satisfiability problem for Linear Time Temporal Logic LTL, CTL and PDL in Chapter 6. They lead to a polynomial space decision procedure for LTL, and exponential time decision procedures for CTL and PDL. Here, winning strategies are only history-free for the existential player. The universal player s strategies depend on a finite part of the history of a play. In spite of the strong connections between tableaux, automata and games their differences are more than simply a matter of taste. Complete axiomatisations for LTL, CTL and PDL can be extracted from the satisfiability focus games in an elegant way. This is done in Chapter 7 by formulating the game rules, the winning conditions and the winning strategies in terms of an axiom system. Completeness of this system then follows from the fact that the existential player wins the game on a consistent formula, i.e. it is satisfiable. We also introduce satisfiability games for CTL* based on the focus approach. They lead to a double exponential time decision procedure. As in the LTL, CTL and PDL case, only the existential player has history-free winning strategies. Since these strategies witness satisfiability of a formula and stay in close relation to its syntactical structure, it might be possible to derive a complete axiomatisation for CTL* from these games as well. Finally, Chapter 9 deals with Fixed Point Logic with Chop, FLC. It extends modal µ-calculus with a sequential composition operator. Satisfiability for FLC is undecidable but its model checking problem remains decidable. In fact it is hard for polynomial space. We give two different game-based solutions to the model checking problem for FLC. Deciding the winner for both types of games meets this polynomial space lower bound for formulas with fixed alternation (and sequential) depth. In the general case the winner can be determined using exponential time, resp. exponential space. The former result holds for games that give rise to global model checking whereas the latter describes the complexity of local FLC model checking. FLC is interesting for verification purposes since it --- unlike all the other logics discussed here --– can describe properties which are non-regular. The thesis concludes with remarks and comments on further research in the area of games for modal and temporal logics

Games for the Strategic Influence of Expectations

We introduce a new class of games where each player's aim is to randomise her strategic choices in order to affect the other players' expectations aside from her own. The way each player intends to exert this influence is expressed through a Boolean combination of polynomial equalities and inequalities with rational coefficients. We offer a logical representation of these games as well as a computational study of the existence of equilibria.Comment: In Proceedings SR 2014, arXiv:1404.041